school on synchrotron and fel based methods and their multi- disciplinary...

2332-6

School on Synchrotron and FEL Based Methods and their Multi-Disciplinary Applications

Peter Laggner

19 - 30 March 2012

University of Graz (Austria)

Small-Angle X-Ray Scattering � Fundamentals and Applications

19.03.2012

1

Small-Angle X-Ray Scattering –Fundamentals and Applications

Peter Laggner, Heinz Amenitsch, Michael Rappolt, Georg Pabst, Maria Schmuck, Benedetta Marmirolli, and Manfred Kriechbaum

ICTP School Trieste 210312March 19, 2012 1

CONTENTS

I. What is SAXS ?

II. General Areas of Application

III. Some Basic Physics

IV. Key Examples

V. Advanced Techniques / Synchrotron / Home-Laboratory


SAXNanoparticle Size and Shape,

10 – 1000 Å

WAXInner MolecularStructure

< 10Å

X-rays

< 5

> 5


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March 19, 2012 ICTP School Trieste 210312 4

The length scalesA Note on Scales

nm µm mm

Microporous Hydroxyapatite

This is where all the chemistry happens

Liposomal Carrier

The Product

This is our reality

x 106

SWAXS

log t (s)-6 -4 -2 0 2

Synchrotron beam-lines Lab-instruments

Typical Time-Scales

Small – Angle : Nano-Domain EnvelopeSize/Shape

The Reciprocity : Scattering Angle – Real Space Dimension

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SWAXS - simultaneous SAXS and WAXS

SAXS

1150 nm 10 3Angström

WAXScrystalline

amorphous

Nanometric density fluctuations

Nanoscale pores/ particles

Size - Shape

Atomic / molecularorder/disorder

Molecular packing

A blind-man‘s approach to SAXS:

Curve width: (reciprocal) size of inhomogeneities

Final slope: inner surface

Scattering angle

Inte

nsi

ty

Integral SAXS: domain volume


Specific inner surface, pore size

SAXS

WAXS Molecular crystal structure

SWAXS – simultaneous small-and wide angle scattering


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SAXS

WAXS

Molecular LatticeUnit cell dimensionssymmetry

Nano-EnvelopeSize / Shape

The angular resolution needs to be much better for SAXS than for XRD –

only ~ 6° ROI for two decades in real space

Why is SAXS special ?

ICTP School Trieste 210312

Nanomaterials

Structural Biology

Pharmacology

BiomaterialsNanocomposites

SWAXS Applications

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Discovery Development Production Stability

•Proteomics

•HTP-screening

•Polymorph screening

•Hetero-Phase diagrams

•Controlled release

•QC

•PAT

•Aging

•Phase transitions

•Crystallization

Where can SAXS Provide Solutions for Chemistry ?

all these areasneed fast laboratory SAXS

Example Pharmaceutical Products


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SAXS – wherever nanostructure counts

Bulk powders

Liquid crystals

Biomaterials Polymers

Nanomaterials Proteins

SynchrotronLaboratory

Industry

Extreme conditions:

• fast kinetics – molecular movie

• thin film structure – GISAXS

• microfluidics

• micro- and nanobeams

• gas phase

• integration with other techniques

Tools for R&D, QC, PAT:

• constant availability

• speed and economy

• reliable, robust, automatisation

• on-line process control

• lab-GISAXS

• integration with other techniques

The main routes of development:

New techniques

New problems

The Routes of Innovation


The SAXS Experiment

source &monochromator

pinholes

sample

beam-stop

area-detector

c

Resolution1 - 100 nm

Other definitions:


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SAXS Theory

Fourier-integral

neutrons: b = nuclear scattering length(magnetic formfactor)

X-rays: b = atomic formfactor fZ

SAS: length scale >> atomic distances

Continuous scattering–length density


Two-phase Approximation

Two homogeneous phases ( and ) with sharp interfaces

Isotropic systems: spherical average

For isotropic systems (fluids, glasses, polycrystals)

no direction dependence of the scattered radiation


Two general laws

Integrated Intensity = volume fraction

Porods Law

S = total interface

per unit volume

Porod radiuse.g. <<1 and spherical particles

Rp = (/3) R R


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The four limiting cases

Dilute particles Crowded particles

Random porous/2-phase Liquid crystalline


Radius of gyration, particlesize, shape

Solution structure factor

Specific inner surface, pore size

Lattice dimensions, unit cell structure


Examples:

•Protein solutions

•Polymer solutions

•Nanoparticle solutions

Parameters

•Radius of Gyration

•Particle weight

•Particle Volume

Particle Shape

Dilute, Monodisperse System


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Rg = 15.5 Å

Solution SAX-Scattering of Lysozyme in 2% aqueous

Buffer:X-ray power: 2kW (CuKα), exposure-time: 1000 s

Background-subtracted raw-data of lysozyme (2%)

Guinier-Plot

q (Å-1)

Inte

nsi

ty (

cou

nts

)

q2



Rapid Particle Sizing

lysozyme (2%)

q (Å-1)

Inte

nsi

ty (

cou

nts

)

q1/2 ~ 0.12 Å-1

Radius ~ π/ q1/2 ~ 26 Å

Poor-man‘s approach

March 19, 2012 23

The Pair Density Distribution Function p(R)

For globular particles:

Possible to determine :

maximum dimension of the particle

radius of gyration

zero-angle intensity

different particle geometries


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Crowded systemq

I

Experimental curve

S

q

P

q

Solution Structure Factor Particle Form Factor

x


Example:Particle Interaction

Scattering data are normalized to their concentration


SAXS is the most universal method for nanostructure analysis:

•Sensitive – low concentration (down to 0.1 %)

•Noninvasive – no preparation, staining or drying

•In-situ, real-time capabilities

•Extremely versatile (solids, liquids, gas phase)

The problem is how to retrieve the structural information from the scatteringpatterns

Principle of Solid-State SAXS


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The thermodynamic stability of a heterogeneous system (e.g. crystalline/amorphous) is not primarily determined by the percentage of the phases but by the interface per unit volume.

1 % amorphous can be harmless, or disatrous – depending on Si

The Solid-State Problem: Inner Surface

Lower inner surface

Higher stability

Higher inner surface

Lower stability

Equal %amorphous

or

porosity


SAXS

XRPD

µm


SAXS : A Measure of Stability

SAXS provides a measure for the specific inner surface(area per volume) between domains of different density

Density differences can arise from• Crystalline/amorphous domain structure• nanoporesBoth are thermodynamic instabilities: Gibbs‘ free

energy grows with inner surface area - the larger the inner surface, the lower the stability


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Recalling an old principle of small-angle scattering:

Porod, G. , Kolloid-Z. 124, 83 (1951)

l1

l2

l3

l4


l1

l2

l3

l4

The “Invariant“ Q:

The integral scattering, the ‚Invariant‘ , is equal to the total irradiated volume times the mean-square electron density

fluctuation – independent of domain shape. (Debye, Bueche)

Random porous/2-phase System


GENERAL RELATIONS (2)

In the case of a two-phase system ( e.g. crystalline/ amorphous polymer), the invariant is related to the

volume fractions , and the electron densities c and a

total irradiated volume


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GENERAL RELATIONS (3)

The decay constant k from a two-phase system is given by

K depends on the total inner surface and the mean-sqare electron density

fluctuations



l1

l2

l3

l4

The inverse of k/Q is equalto the mean chordlength:

The mean pore diameters and wall diameters can be calculated from the relations:

and

March 19, 2012 35

How to obtain Si from experimental data:

• use the relation:

determine Porod limits from log I /log q –plot.

The slope should be –4 (at least close to that), according to I~q-4. If far from –4, system isfractal or has transition layer, and Si cannot be determined.

• determine K, from extrapolation to lg q = 0

• determine Q from integration (extrapolation to zero and infinity)

• Calculate Si in units of m2/g from :

p ... volume fraction of pores (e.g. He-pycnometry)

and s ... matrix density of the solid (e.g. crystal data).


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Comparison SAXS/BET

(BAM – Alumina Standards)

SAXS inner surface is higher than BET:

• inaccessible, closed pores

• nanostructured particle surface

Both are equally relevant for stability

Why?

BETm2/g

SAXSm2/g

PM-102 5,41 6,1

PM-104 79,8 97,4

PM-103 156,0 161,4


Inner surface by SAXS

... within minutes

....without drying

Mesoporous Materials - MCM

Nanomaterials

M.Lüchinger, G.Pirngruber, B.Lidlar, R.Prins and P. Laggner, J.Chem.Phys. (2004)


Monitoring Shape and Size of Pores in M41S Silicas

M.Lüchinger, G.Pirngruber, B.Lidlar, R.Prins and P. Laggner (2004)

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Synthesis procedure

dissolved silica (sodium silicate solution)

+ quaternary ammonium detergent micelles

organic, inorganic mesophase

mesoporous silica

100°C, 96 h 540°C in Air

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